64 research outputs found
Subshifts, MSO Logic, and Collapsing Hierarchies
We use monadic second-order logic to define two-dimensional subshifts, or
sets of colorings of the infinite plane. We present a natural family of
quantifier alternation hierarchies, and show that they all collapse to the
third level. In particular, this solves an open problem of [Jeandel & Theyssier
2013]. The results are in stark contrast with picture languages, where such
hierarchies are usually infinite.Comment: 12 pages, 5 figures. To appear in conference proceedings of TCS 2014,
published by Springe
On the Expressive Power of 2-Stack Visibly Pushdown Automata
Visibly pushdown automata are input-driven pushdown automata that recognize
some non-regular context-free languages while preserving the nice closure and
decidability properties of finite automata. Visibly pushdown automata with
multiple stacks have been considered recently by La Torre, Madhusudan, and
Parlato, who exploit the concept of visibility further to obtain a rich
automata class that can even express properties beyond the class of
context-free languages. At the same time, their automata are closed under
boolean operations, have a decidable emptiness and inclusion problem, and enjoy
a logical characterization in terms of a monadic second-order logic over words
with an additional nesting structure. These results require a restricted
version of visibly pushdown automata with multiple stacks whose behavior can be
split up into a fixed number of phases. In this paper, we consider 2-stack
visibly pushdown automata (i.e., visibly pushdown automata with two stacks) in
their unrestricted form. We show that they are expressively equivalent to the
existential fragment of monadic second-order logic. Furthermore, it turns out
that monadic second-order quantifier alternation forms an infinite hierarchy
wrt words with multiple nestings. Combining these results, we conclude that
2-stack visibly pushdown automata are not closed under complementation.
Finally, we discuss the expressive power of B\"{u}chi 2-stack visibly pushdown
automata running on infinite (nested) words. Extending the logic by an infinity
quantifier, we can likewise establish equivalence to existential monadic
second-order logic
Automata with Nested Pebbles Capture First-Order Logic with Transitive Closure
String languages recognizable in (deterministic) log-space are characterized
either by two-way (deterministic) multi-head automata, or following Immerman,
by first-order logic with (deterministic) transitive closure. Here we elaborate
this result, and match the number of heads to the arity of the transitive
closure. More precisely, first-order logic with k-ary deterministic transitive
closure has the same power as deterministic automata walking on their input
with k heads, additionally using a finite set of nested pebbles. This result is
valid for strings, ordered trees, and in general for families of graphs having
a fixed automaton that can be used to traverse the nodes of each of the graphs
in the family. Other examples of such families are grids, toruses, and
rectangular mazes. For nondeterministic automata, the logic is restricted to
positive occurrences of transitive closure.
The special case of k=1 for trees, shows that single-head deterministic
tree-walking automata with nested pebbles are characterized by first-order
logic with unary deterministic transitive closure. This refines our earlier
result that placed these automata between first-order and monadic second-order
logic on trees.Comment: Paper for Logical Methods in Computer Science, 27 pages, 1 figur
Quantified CTL: Expressiveness and Complexity
While it was defined long ago, the extension of CTL with quantification over
atomic propositions has never been studied extensively. Considering two
different semantics (depending whether propositional quantification refers to
the Kripke structure or to its unwinding tree), we study its expressiveness
(showing in particular that QCTL coincides with Monadic Second-Order Logic for
both semantics) and characterise the complexity of its model-checking and
satisfiability problems, depending on the number of nested propositional
quantifiers (showing that the structure semantics populates the polynomial
hierarchy while the tree semantics populates the exponential hierarchy)
Modal Fragments of Second-Order Logic
Formaalin logiikan tutkimuskohteina ovat erilaiset muodolliset systeemit eli logiikat, joiden avulla voidaan mm. mekanisoida monenlaisia päättelyprosesseja. Eräs modernin formaalin logiikan keskeisistä tutkimusaiheista on modaalilogiikka, jossa perinteisempää logiikkaa laajennetaan nk. modaliteeteilla. Modaliteettien avulla voidaan luoda mitä erilaisimpia formaaleja systeemejä. Modaalilogiikalla onkin huomattava määrä sovelluksia aina tietojenkäsittelytieteestä ja matematiikan sekä fysiikan perusteista filosofiaan ja kielitieteisiin.
Väitöskirja keskittyy modaalilogiikan nk. malliteoriaan. Tutkielmassa luokitellaan erilaisia formaalin logiikan systeemejä perustuen siihen, millaisia ominaisuuksia kyseisten systeemien avulla voidaan ilmaista. Mitä korkeampi ilmaisuvoima formaalilla järjestelmällä on, sitä hitaampaa on järjestelmän avulla suoritettava tietokoneellistettu päättely. Tutkielma käsittelee useita modaalilogiikan systeemejä; painopiste on erittäin korkean ilmaisuvoiman omaavien logiikoiden teoriassa. Tarkastelun kohteena olevat kysymykset liittyvät suoraan muuhun modaalilogiikan alan matemaattiseen tutkimukseen. Tutkielmassa mm. esitetään ratkaisu vuodesta 1983 avoinna olleeseen tekniseen kysymykseen koskien nk. toisen kertaluvun propositionaalisen modaalilogiikan alternaatiohierarkiaa.In this thesis we investigate various fragments of second-order logic that arise naturally in considerations related to modal logic. The focus is on questions related to expressive power. The results in the thesis are reported in four independent but related chapters (Chapters 2, 3, 4 and 5).
In Chapter 2 we study second-order propositional modal logic, which is the system obtained by extending ordinary modal logic with second-order quantification of proposition symbols. We show that the alternation hierarchy of this logic is infinite, thereby solving an open problem from the related literature. In Chapter 3 we investigate the expressivity of a range of modal logics extended with existential prenex quantification of accessibility relations and proposition symbols. The principal result of the chapter is that the resulting extension of (a version of) Boolean modal logic can be effectively translated into existential monadic second-order logic. As a corollary we obtain decidability results for multimodal logics over various classes of frames with built-in relations.
In Chapter 4 we study the equality-free fragment of existential second-order logic with second-order quantification of function symbols. We show that over directed graphs, the expressivity of the fragment is incomparable with that of first-order logic. We also show that over finite models with a unary relational vocabulary, the fragment is weaker in expressivity than first-order logic.
In Chapter 5 we study the extension of polyadic modal logic with unrestricted quantification of accessibility relations and proposition symbols. We obtain a range of results related to various natural fragments of the system. Finally, we establish that this extension of modal logic exactly captures the expressivity of second-order logic
- …